The lines which bisect the area of a triangle form an envelope as shown in this picture

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It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is $$\frac{3}{4} \log_e(2) - \frac{1}{2} \approx 0.01986.$$

But this is also $$\sum_{n=1}^{\infty}\frac{1}{(4n-1)(4n)(4n+1)}.$$

Is there any connection between the series and the deltoid? Or is it just a coincidence?



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